Tightness and weak convergence in the topology of local uniform convergence for stochastic processes in the dual of a nuclear space

Abstract

Let ${\Phi }^{\prime }$ denote the strong dual of a nuclear space $\Phi$ and let $C_{\infty }({\Phi }^{\prime })$ be the collection of all continuous mappings $x:[0,\infty )\to {\Phi }^{\prime }$ equipped with the topology of local uniform convergence. In this paper we prove sufficient conditions for tightness of probability measures on $C_{\infty }({\Phi }^{\prime })$ and for weak convergence in $C_{\infty }({\Phi }^{\prime })$ for a sequence of ${\Phi }^{\prime }$-valued processes. We illustrate our results with two applications. First, we show the central limit theorem for local martingales taking values in the dual of an ultrabornological nuclear space. Second, we prove sufficient conditions for the weak convergence in $C_{\infty }({\Phi }^{\prime })$ for a sequence of solutions to stochastic partial differential equations driven by semimartingale noise.

The original submission has been split into two separated works. Tightness and weak uniform convergence covered in this work, and UCP convergence is studied in the separated work arXiv:2303.17082

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